Toeplitz operators with distributional symbols on Bergman spaces

We study the boundedness and compactness of Toeplitz operators Ta on Bergman spaces , 1 < p < ∞. The novelty is that we allow distributional symbols. It turns out that the belonging of the symbol to a weighted Sobolev space of negative order is sufficient for the boundedness of Ta. We show the natural relation of the hyperbolic geometry of the disc and the order of the distribution. A corresponding sufficient condition for the compactness is also derived.

A note on the Fredholm properties of Toeplitz operators on weighted Bergman spaces with matrix-valued symbols

We characterize the essential spectra of Toeplitz operators Ta on weighted Bergman spaces with matrix-valued symbols; in particular we deal with two classes of symbols, the Douglas algebra C+H∞ and the Zhu class Q := L∞ ∩VMO∂ . In addition, for symbols in C+H∞ , we derive a formula for the index of Ta in terms of its symbol a in the scalar-valued case, while in the matrix-valued case we indicate that the standard reduction to the scalar-valued case fails to work analogously to the Hardy space case. Mathematics subject classification (2010): 47B35,

New results and open problems on Toeplitz operators in Bergman spaces

We discuss some of the recent progress in the field of Toeplitz operators acting on Bergman spaces of the unit disk, formulate some new results, and describe a list of open problems -- concerning boundedness, compactness and Fredholm properties -- which was presented at the conference "Recent Advances in Function Related Operator Theory'' in Puerto Rico in March 2010.

Toeplitz operators on Bergman spaces with locally integrable symbols

We study the boundedness of Toeplitz operators $T_a$ with locally integrable symbols on Bergman spaces $A^p(\mathbb{D})$, $1 < p < \infty$. Our main result gives a sufficient condition for the boundedness of $T_a$ in terms of some ``averages'' (related to hyperbolic rectangles) of its symbol. If the averages satisfy an ${o}$-type condition on the boundary of $\mathbb{D}$, we show that the corresponding Toeplitz operator is compact on $A^p$. Both conditions coincide with the known necessary conditions in the case of nonnegative symbols and $p=2$. We also show that Toeplitz operators with symbols of vanishing mean oscillation are Fredholm on $A^p$ provided that the averages are bounded away from zero, and derive an index formula for these operators.

Hankel operators on Fock spaces

We study Hankel operators on the weighted Fock spaces Fp. The boundedness and compactness of these operators are characterized in terms of BMO and VMO, respectively. Along the way, we also study Berezin transform and harmonic conjugates on the plane. Our results are analogous to Zhu's characterization of bounded and compact Hankel operators on Bergman spaces of the unit disk.

Universally reversible JC*-triples and operator spaces

Abstract. We prove that the vast majority of JC∗-triples satisfy the condition of universal reversibility. Our characterisation is that a JC∗-triple is universally reversible if and only if it has no triple homomorphisms onto Hilbert spaces of dimension greater than two nor onto spin factors of dimension greater than four. We establish corresponding characterisations in the cases of JW∗-triples and of TROs (regarded as JC∗-triples). We show that the distinct natural operator space structures on a universally reversible JC∗-triple E are in bijective correspondence with a distinguished class of ideals in its universal TRO, identify the Shilov boundaries of these operator spaces and prove that E has a unique natural operator space structure precisely when E contains no ideal isometric to a nonabelian TRO. We deduce some decomposition and completely contractive properties of triple homomorphisms on TROs.

Contested institutions? Traditional leaders and land access and control in communal areas of Eastern Cape Province, South Africa

The South African government has endeavoured to strengthen property rights in communal areas and develop civil society institutions for community-led development and natural resource management. However, the effectiveness of this remains unclear as the emergence and operation of civil society institutions in these areas is potentially constrained by the persistence of traditional authorities. Focusing on the former Transkei region of Eastern Cape Province, three case study communities are used examine the extent to which local institutions overlap in issues of land access and control. Within these communities, traditional leaders (chiefs and headmen) continue to exercise complete and sole authority over land allocation and use this to entrench their own positions. However, in the absence of effective state support, traditional authorities have only limited power over how land is used and in enforcing land rights, particularly over communal resources such as rangeland. This diminishes their local legitimacy and encourages some groups to contest their authority by cutting fences, ignoring collective grazing decisions and refusing to pay ‘fees’ levied on them. They are encouraged in such activities by the presence of democratically elected local civil society institutions such as ward councillors and farmers’ organisations, which have broad appeal and are increasingly responsible for much of the agrarian development that takes place, despite having no direct mandate over land. Where it occurs at all, interaction between these different institutions is generally restricted to approval being required from traditional leaders for land allocated to development projects. On this basis it is argued that a more radical approach to land reform in communal areas is required, which transfers all powers over land to elected and accountable local institutions and integrates land allocation, land management and agrarian development more effectively.

Rigid body trajectories in different 6D spaces

The objective of this paper is to show that the group SE(3) with an imposed Lie-Poisson structure can be used to determine the trajectory in a spatial frame of a rigid body in Euclidean space. Identical results for the trajectory are obtained in spherical and hyperbolic space by scaling the linear displacements appropriately since the influence of the moments of inertia on the trajectories tends to zero as the scaling factor increases. The semidirect product of the linear and rotational motions gives the trajectory from a body frame perspective. It is shown that this cannot be used to determine the trajectory in the spatial frame. The body frame trajectory is thus independent of the velocity coupling. In addition, it is shown that the analysis can be greatly simplified by aligning the axes of the spatial frame with the axis of symmetry which is unchanging for a natural system with no forces and rotation about an axis of symmetry.

Solvability and spectral properties of integral equations on the real line: I. Weighted spaces of continuous functions

We consider in this paper the solvability of linear integral equations on the real line, in operator form (λ−K)φ=ψ, where and K is an integral operator. We impose conditions on the kernel, k, of K which ensure that K is bounded as an operator on . Let Xa denote the weighted space as |s|→∞}. Our first result is that if, additionally, |k(s,t)|⩽κ(s−t), with and κ(s)=O(|s|−b) as |s|→∞, for some b>1, then the spectrum of K is the same on Xa as on X, for 01. As an example where kernels of this latter form occur we discuss a boundary integral equation formulation of an impedance boundary value problem for the Helmholtz equation in a half-plane.

Fractal spaces in planning and governance

This paper discusses concepts of space within the planning literature, the issues they give rise to and the gaps they reveal. It then introduces the notion of 'fractals' borrowed from complexity theory and illustrates how it unconsciously appears in planning practice. It then moves on to abstract the core dynamics through which fractals can be consciously applied and illustrates their working through a reinterpretation of the People's Planning Campaign of Kerala, India. Finally it highlights the key contribution of the fractal concept and the advantages that this conceptualisation brings to planning.

Conspicuous redemption? Reflections on the promises and perils of the 'celebritization' of climate change

With rising public awareness of climate change, celebrities have become an increasingly important community of non nation-state ‘actors’ influencing discourse and action, thereby comprising an emergent climate science–policy–celebrity complex. Some feel that these amplified and prominent voices contribute to greater public understanding of climate change science, as well as potentially catalyze climate policy cooperation. However, critics posit that increased involvement from the entertainment industry has not served to influence substantive long-term advancements in these arenas; rather, it has instead reduced the politics of climate change to the domain of fashion and fad, devoid of political and public saliency. Through tracking media coverage in Australia, Canada, the United States, and United Kingdom, we map out the terrain of a ‘Politicized Celebrity System’ in attempts to cut through dualistic characterizations of celebrity involvement in politics. We develop a classification system of the various types of climate change celebrity activities, and situate movements in contemporary consumer- and spectacle-driven carbon-based society. Through these analyses, we place dynamic and contested interactions in a spatially and temporally-sensitive ‘Cultural Circuits of Climate Change Celebrities’ model. In so doing, first we explore how these newly ‘authorized’ speakers and ‘experts’ might open up spaces in the public sphere and the science/policy nexus through ‘celebritization’ effects. Second, we examine how the celebrity as the ‘heroic individual’ seeking ‘conspicuous redemption’ may focus climate change actions through individualist frames. Overall, this paper explores potential promises, pitfalls and contradictions of this increasingly entrenched set of ‘agents’ in the cultural politics of climate change. Thus, as a form of climate change action, we consider whether it is more effective to ‘plant’ celebrities instead of trees.

Sparse polynomial approximation in positive order Sobolev spaces with bounded mixed derivatives and applications to elliptic problems with random loading

In the present paper we study the approximation of functions with bounded mixed derivatives by sparse tensor product polynomials in positive order tensor product Sobolev spaces. We introduce a new sparse polynomial approximation operator which exhibits optimal convergence properties in L2 and tensorized View the MathML source simultaneously on a standard k-dimensional cube. In the special case k=2 the suggested approximation operator is also optimal in L2 and tensorized H1 (without essential boundary conditions). This allows to construct an optimal sparse p-version FEM with sparse piecewise continuous polynomial splines, reducing the number of unknowns from O(p2), needed for the full tensor product computation, to View the MathML source, required for the suggested sparse technique, preserving the same optimal convergence rate in terms of p. We apply this result to an elliptic differential equation and an elliptic integral equation with random loading and compute the covariances of the solutions with View the MathML source unknowns. Several numerical examples support the theoretical estimates.